The
nonabelian tensor square G[otimes ] G of a group G is generated by the symbols
g[otimes ] h, g,h ∈ G, subject to the relations
$$gg\prime\otimesh=(^gg\prime\otimes^gh)(g\otimesh) and
g\otimeshh\prime-(g\otimesh)(^hg\otimes^hh\prime),$$ for all $g,g\prime,h,h\prime
\in G< / f>, where $^gg\prime=gg\primeg^{−1}$. The nonabelian tensor square
is a special case of the nonabelian tensor product which has its origins in
homotopy theory. It was introduced by R. Brown and J.-L. Loday in [4]
and [5], extending ideas of J.H.C. Whitehead in [10]. The topic
of this paper is the classification of 2-generator 2-groups of class two up to
isomorphism and the determination of nonabelian tensor squares for these
groups.